When using a coordinate measuring machine (CMM) to ascertain, say, circularity
(or sphericity) within a prescribed tolerance, one needs to find a suitable
annulus, that is, the region inside the larger and outside the smaller of two
concentric circles that enclose the measured data points. If the distance
between the bounding circles exceeds tolerance, then the circularity test has
failed. A linear-programming approach to finding the smallest annulus is
proposed.

Much of the Nation's manufacturing industry depends on the availability of
accurate coordinate measurements. The Mechanical Engineering Laboratory of NIST
has been pivotal in examining CMM performance and in issuing guidelines for
their proper use. ACMD can contribute to this effort through its expertise in
optimization and computational geometry techniques.

Finding the minimum annulus is a nonlinear-programming problem. It is not
convex so that local minima need not be global. A complicated
computational-geometric procedure avoids that difficulty but requires the
examination of all intersections of the closest and farthest point Voronoi
diagrams, respectively, of the data points.

The minimum annulus problem requires finding the circle which best fits given
points in the sense of Chebychev: minimize the maximum orthogonal distance from
the circle. The Chebychev necessary optimality condition for an enclosing
annulus requires that four of the data points occupy the bounding circles,
limiting potential solutions to a finite set.

The same necessary optimality condition holds for a slightly modified problem:
minimize the maximum squared orthogonal distance. This problem turns out to
have a straightforward linear-programming solution which obviates the
examination of local minima. While examples can be constructed in which the
actual minimum annulus and the ``squared" minimum annulus differ, they coincide
in most observed instances derived from actual measurements. Analogous results
were found for spherical fits. At the INFORMS conference in Washington, DC, on
May 8, the squared minimum annulus was therefore proposed as an excellent
approximation to, and likely to coincide with, the actual minimum annulus.

Plans are to search for conditions under which the squared minimum annulus
coincides with the actual minimum annulus, and develop specialized
linear-programming software.